2.1 The physical domain

Following the treatment of Watson (Reference Watson1964) of the flow over a stationary disk, we identify four distinct flow regions for the jet over a rotating disk, with smooth passage from one region to the next (see figure 1): a stagnation flow region (i), a developing boundary-layer region (ii), where the boundary layer grows until it reaches the film surface at $r=r_{0}$ , and a fully viscous thin-film region (iii). Under some flow conditions, a hydraulic jump may form in region (iv), starting at $r=r_{j}$ . In the vicinity of the stagnation point, in region (i), $r=O(1)$ . The velocity outside the boundary layer rises rapidly from 0 at the stagnation point to the impingement velocity in the inviscid far region. In region (ii), the boundary layer is expected to grow like $\sqrt{r/Re}$ , at a rate that diminishes with $\unicode[STIX]{x1D6FA}$ . The elongation rate or normal stress weakens with distance, but the flow remains axisymmetric (as opposed to becoming almost planar) with non-negligible surface curvature. The speed outside the boundary layer remains almost constant, equal to 1 (in units of $Q/\unicode[STIX]{x03C0}a^{2}$ ), as the fluid here is unaffected by the viscous stresses. For $r\gg 1$ , the flow field in region (ii) is not significantly affected by the stagnation flow of region (i). The region $1\ll r<r_{0}$ will be referred to as the developing boundary-layer region, with boundary-layer thickness $\unicode[STIX]{x1D6FF}(r)$ , outside which the flow is inviscid and constant. Here $r_{0}$ is the location of the transition point at which viscous stresses become appreciable right up to the free surface, where the whole flow is of the boundary-layer type. At this point, the velocity profile changes from the Blasius type to the similarity profile for a stationary disk. In contrast, a similarity profile does not exist for a jet impinging on a rotating disk. The flow in region (iii), $r>r_{0}$ , which will be referred to as the fully developed viscous region, is bounded by the disk and the free surface $z=h(r)$ .

Finally, the hydraulic jump in region (iv) occurs at a location $r=r_{j}$ , which can be larger or smaller than $r_{0}$ since the jump may occur at any point in the flow. It is convenient to introduce the (supercritical) film thickness upstream of the jump as $h(r)=h(r<r_{j})$ and the (subcritical) thickness downstream of the jump as $H(r)\equiv h(r>r_{j})$ . The jump height $H(r)$ is generally not constant but is different from the height immediately downstream of the jump, $H_{j}=H(r=r_{j})$ . In this study, the fluid is assumed to be drained at the edge of the disk $r=r_{\infty }$ to maintain steady flow, with the film thickness denoted by $H_{\infty }=H(r=r_{\infty })$ . Although it is common practice to assume the jump height to remain equal to $H_{j}$ , this assumption is valid for fluids of low viscosity on a stationary disk. The edge thickness is expected to depend heavily on the rotation speed and surface tension, but not so much on the flow rate (Mohajer & Li Reference Mohajer and Li2015). In this study, we consider a viscous layer downstream of the jump.

2.2 Governing equations and boundary conditions

Unless otherwise specified, the Reynolds number is assumed to be moderately large so that our analysis is confined to the laminar regime. Consequently, for steady axisymmetric thin-film flow, in the presence of rotation, the mass and momentum conservation equations are formulated using a thin-film or Prandtl boundary-layer approach, which amounts to assuming that the radial flow varies much slower than the vertical flow (Schlichtling Reference Schlichtling2000). The thin-film problem is formulated in the appendix. By letting a subscript with respect to $r$ or $z$ denote partial differentiation, the reduced dimensionless conservation equations become

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{r}+{\displaystyle \frac{u}{r}}+w_{z}=0, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle Re\left(uu_{r}+wu_{z}-{\displaystyle \frac{v^{2}}{r}}\right)=-{\displaystyle \frac{Re}{Fr^{2}}}p_{r}+u_{zz}, & \displaystyle\end{eqnarray}$$

(2.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle Re\left(uv_{r}+wv_{z}+{\displaystyle \frac{uv}{r}}\right)=v_{zz}, & \displaystyle\end{eqnarray}$$

(2.1d ) $$\begin{eqnarray}\displaystyle & \displaystyle p_{z}=\left\{\begin{array}{@{}ll@{}}0\quad & r<r_{j}\\ -1\quad & r>r_{j}.\end{array}\right. & \displaystyle\end{eqnarray}$$

These equations are essentially the same as those used by Bohr et al. (Reference Bohr, Ellegaard, Hansen and Haaning1996). We observe that the pressure for a thin film is essentially hydrostatic as a result of its vanishing at the film surface (in the absence of surface tension) and the small thickness of the film. In addition, upstream of the jump, the variation of the film thickness with the radius is expected to be smooth and gradual. In this case, the radial variation of the hydrostatic pressure is also small. According to the calculations of Prince et al. (Reference Prince, Maynes and Crockett2012), the hydrostatic pressure exerts less than 0.4 % cumulative influence on the dynamics of the thin film, and is thus neglected upstream of the hydraulic jump. This is generally commonly assumed in modelling hydraulic jump flow, where the hydrostatic pressure is included only downstream of the jump where the film is relatively thicker (Watson Reference Watson1964; Bush & Aristoff Reference Bush and Aristoff2003; Dressaire et al. Reference Dressaire, Courbin, Crest and Stone2010; Prince et al. Reference Prince, Maynes and Crockett2012). At the disk, the no-slip and no-penetration conditions are assumed to hold for any $r$ . In this case:

(2.2a-c ) $$\begin{eqnarray}u(r,z=0)=0,\quad v(r,z=0)=\unicode[STIX]{x1D6FA}r,\quad w(r,z=0)=0.\end{eqnarray}$$

At the free surface $z=h(r<r_{j})$ or $z=H(r>r_{j})$ , the kinematic condition for steady flow takes the form

(2.3) $$\begin{eqnarray}w(r,z=h)=u(r,z=h)h^{\prime }(r).\end{eqnarray}$$

Here a prime denotes total differentiation. In addition, the radial and tangential shear stress components as well as the pressure must vanish everywhere $(0<r<r_{\infty })$ , so that

(2.4a-c ) $$\begin{eqnarray}u_{z}(r,z=h)=v_{z}(r,z=h)=p(r,z=h)=0.\end{eqnarray}$$

The flow field is sought separately in the developing boundary-layer region (ii) for $0<r<r_{0}$ , the fully developed viscous boundary-layer region (iii) for $r_{0}<r<r_{j}$ and the hydraulic jump region (iv) for $r_{j}<r<r_{\infty }$ . We observe that region (i) is negligible, and the leading edge of the boundary layer in region (ii) coincides with the disk centre (see § 2.3). Consequently, the additional boundary conditions are as follows. In region (ii), the flow is assumed to be sufficiently inertial for inviscid flow to prevail between the boundary-layer outer edge and the free surface (see figure 1). In this case, the following condition at the outer edge of the boundary layer and beyond must hold:

(2.5a-c ) $$\begin{eqnarray}u(r<r_{0},z=\unicode[STIX]{x1D6FF})=1,\quad u_{z}(r<r_{0},z=\unicode[STIX]{x1D6FF})=0,\quad u(r<r_{0},\unicode[STIX]{x1D6FF}\leqslant z<h)=1.\end{eqnarray}$$

Integrating equation (2.1d ) subject to condition (2.4c ), the pressure becomes

(2.6) $$\begin{eqnarray}p(r,z)=\left\{\begin{array}{@{}ll@{}}0\quad & r<r_{j}\\ H(r)-z\quad & r\geqslant r_{j}.\end{array}\right.\end{eqnarray}$$

Equations (2.1b ) and (2.1c ) must be satisfied at the rotating disk. Thus, we have

(2.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{zz}(r,z=0)=-Re\unicode[STIX]{x1D6FA}^{2}r+\left\{\begin{array}{@{}ll@{}}0\quad & r<r_{j}\\ G{\displaystyle \frac{\text{d}H}{\text{d}r}}\quad & r>r_{j},\end{array}\right. & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle v_{zz}(r,z=0)=0, & \displaystyle\end{eqnarray}$$

$u(r,z)$

$w(r,z)$

(2.8) $$\begin{eqnarray}v(r,z)=\unicode[STIX]{x1D6FA}r.\end{eqnarray}$$

This simple behaviour is not unexpected for a thin film flowing with a free surface and adhering to the rotating disk. We note that (2.8) holds throughout the flow domain. Finally, the conservation of mass at any location upstream and downstream of the jump yields the following relation in dimensionless form:

(2.9) $$\begin{eqnarray}{\displaystyle \frac{1}{2r}}=\int _{0}^{h(r)}u(r<r_{j},z)\,\,\text{d}z=\int _{0}^{H(r)}u(r>r_{j},z)\,\,\text{d}z.\end{eqnarray}$$

We note that this relation follows from the mass conservation relation in dimensional form: $Q=2\unicode[STIX]{x03C0}\bar{r}\int _{0}^{\bar{h}(\bar{r})}\bar{u}(\bar{r},\bar{z})\,\text{d}\bar{z}$ , where a barred variable denotes a dimensional quantity.

2.3 Overall solution strategy

Throughout this study, the stagnation region (i) under the impinging jet is assumed to be negligibly small. The velocity outside the boundary layer rises rapidly from 0 at the stagnation point to the impingement velocity in the inviscid far region. For a jet on a stationary as well as a rotating disk, the impinging jet is predominantly inviscid close to the stagnation point, and the boundary-layer thickness remains negligibly small. Obviously, this is the case for a jet at relatively large Reynolds number. Disk rotation adds to the inviscid character of the flow upon impingement. Ideally, the flow at the boundary-layer edge should correspond to the (almost parabolic) potential flow near the stagnating jet, with the boundary-layer leading edge coinciding with the stagnation point (Liu, Gabour & Lienhard Reference Liu, Gabour and Lienhard1993). However, the assumption of uniform horizontal flow near the wall and outside the boundary layer (as illustrated in figure 1) is reasonable. The distance from the stagnation point for the inviscid flow to reach uniform horizontal velocity is small, of the order of the jet radius (Lienhard Reference Lienhard2006). In addition, for a jet on a stationary disk, the flow acquires a similarity character. In this case, the position or effect of the leading edge is irrelevant. This is not the case for the current jet on a rotating disk, where, as we shall see, a non-similarity solution is sought subject to initial conditions at the leading edge. However, as argued above, the dominance of inertia near the stagnation point, which is further enhanced by disk rotation, should make plausible the assumption of uniform horizontal flow and negligible stagnation region (i). This assumption, which was adopted initially by Watson (Reference Watson1964), is commonly used in the modelling of impinging jet flow (see, for instance, Higuera Reference Higuera1994; Bush & Aristoff Reference Bush and Aristoff2003; Prince et al. Reference Prince, Maynes and Crockett2012).

The pressure due to gravitational effect is neglected upstream of the hydraulic jump. We shall also revisit the flow of a liquid jet impinging on a stationary disk for reference, for which the similarity solution developed by Watson (Reference Watson1964) applies. In contrast, the rotation of the disk causes the flow to be non-similar in character. Therefore, in the present study, approximate solutions are sought in each region. An integral approach of the Kármán–Pohlhausen type (Schlichtling Reference Schlichtling2000) is adopted upstream of the jump, similar to the formulation of Prince et al. (Reference Prince, Maynes and Crockett2012) for a jet impinging on a slippery disk. The cubic profile may also be considered as the leading-order solution in a comprehensive spectral approach when inertia is included (Khayat & Kim Reference Khayat and Kim2006; Rojas, Argentina & Tirapegui Reference Rojas, Argentina and Tirapegui2010). The cubic profile seems to be amply adequate as it leads to close agreement with Watson’s (Reference Watson1964) similarity solution for a jet impinging on a stationary disk (Prince et al. Reference Prince, Maynes and Crockett2012). The validity of such a profile was also assessed by Khayat (Reference Khayat2016) for a planar jet impinging on a slippery stationary surface; the cubic profile was found to yield good agreement against his numerical solution. See also Rao & Arakeri (Reference Rao and Arakeri1998) for an integral analysis of a rotating film. Higher-order polynomial velocity profiles were also used. In their study on flow separation and wave breaking, Bohr et al. (Reference Bohr, Ellegaard, Hansen and Haaning1996) used a quartic profile to illustrate the emergence of a singularity at the separation point for a thin film. The coefficients in the polynomial expansion were not obtained explicitly. Later, Bohr, Putkaradze & Watanabe (Reference Bohr, Putkaradze and Watanabe1997) adopted a cubic velocity profile in an averaging boundary-layer approach that accounts for regions of separation, yielding the structures of stationary and moving hydraulic jumps. A similar cubic profile will also be adopted for the current problem (see § 4).

Different velocity profiles are used in region (ii) and region (iii) due to the difference in surface velocity. The flow in the entire boundary-layer domain is obtained upon matching the flows at the transition point $r_{0}$ . Downstream of the hydraulic jump, inertia is neglected, but the centrifugal effect is included. However, in contrast to the flow over a stationary disk, an analytical solution is not possible, and the solution will be sought numerically. In an effort to obtain the location and height of the jump uniquely, we impose the value of the film thickness at the edge of the disk. Its value is estimated by applying the minimum energy principle at the disk edge. An additional contribution is included to account for rotation in the absence of radial flow as encountered in spin coating.

3.1 The developing boundary layer in region (ii)

In region (ii), the inviscid flow dominates the upper layer $(\unicode[STIX]{x1D6FF}\leqslant z<h)$ of the film in the radial direction; therefore the radial velocity above the boundary layer remains equal to one. In addition, the tangential component of the inviscid flow outside the boundary-layer edge is negligible near the stagnation point where the Coriolis force is dominated by the impinging flow, which results in a dominant radial flow. A cubic profile is sought for the radial velocity component $u(r,z)$ , which is obtained using conditions (2.2a ), (2.5a,b ) and (2.7a ), that is

(3.1) $$\begin{eqnarray}\displaystyle u(r<r_{0},z)=\left({\displaystyle \frac{3}{2\unicode[STIX]{x1D6FF}}}+{\displaystyle \frac{Re\unicode[STIX]{x1D6FA}^{2}}{4}}r\unicode[STIX]{x1D6FF}\right)z-{\displaystyle \frac{Re\unicode[STIX]{x1D6FA}^{2}}{2}}rz^{2}+\left({\displaystyle \frac{Re\unicode[STIX]{x1D6FA}^{2}}{4}}{\displaystyle \frac{r}{\unicode[STIX]{x1D6FF}}}-{\displaystyle \frac{1}{2\unicode[STIX]{x1D6FF}^{3}}}\right)z^{3}. & & \displaystyle\end{eqnarray}$$

The boundary-layer height $\unicode[STIX]{x1D6FF}$ is determined by considering the mass and momentum balance over the boundary-layer region (ii). Therefore, consider first the integral form of the convective term in (2.1b ). The vertical velocity component is eliminated by noting from (2.1a ) that $w(r,z)=-(1/r)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r)(r\int _{0}^{z}u(r,z)\,\text{d}z)$ . In this case, $uu_{r}+wu_{z}=\unicode[STIX]{x2202}u^{2}/\unicode[STIX]{x2202}r+u^{2}/r-(1/r)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z)(u(r,z)\int _{0}^{z}(\unicode[STIX]{x2202}ru(r,\bar{z})/\unicode[STIX]{x2202}r)\,\text{d}\bar{z})$ . Consequently, upon integrating (2.1b ) over the boundary-layer thickness, and neglecting the radial pressure gradient, we obtain the integral form of the momentum equation upstream of the jump:

(3.2) $$\begin{eqnarray}\left({\displaystyle \frac{\text{d}}{\text{d}r}}+{\displaystyle \frac{1}{r}}\right)\int _{0}^{\unicode[STIX]{x1D6FF}}(u^{2}-u)\,\text{d}z-{\displaystyle \frac{1}{r}}\int _{0}^{\unicode[STIX]{x1D6FF}}v^{2}\,\text{d}z=-{\displaystyle \frac{1}{Re}}u_{z}(r,z=0).\end{eqnarray}$$

Upon substituting the velocity profiles (2.8) and (3.1), we obtain the desired differential equation for the boundary-layer height:

(3.3) $$\begin{eqnarray}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}}{\text{d}r}}=-{\displaystyle \frac{C_{1}r^{2}\unicode[STIX]{x1D6FF}^{6}+C_{2}r\unicode[STIX]{x1D6FF}^{4}+(C_{3}+C_{4}r^{2})\unicode[STIX]{x1D6FF}^{2}+2320r}{C_{5}r^{3}\unicode[STIX]{x1D6FF}^{5}+C_{6}r^{2}\unicode[STIX]{x1D6FF}^{3}+C_{7}r\unicode[STIX]{x1D6FF}}},\end{eqnarray}$$

where the constant coefficients are $C_{1}=3Re^{3}\unicode[STIX]{x1D6FA}^{4},C_{2}=6Re^{2}\unicode[STIX]{x1D6FA}^{2},C_{3}=-234Re$ , $C_{4}=-420(4Re-1)Re\unicode[STIX]{x1D6FA}^{2}$ , $C_{5}=5Re^{3}\unicode[STIX]{x1D6FA}^{4}$ , $C_{6}=9Re^{2}\unicode[STIX]{x1D6FA}^{2}$ and $C_{7}=-234Re$ .

Equation (3.3) is solved numerically subject to $\unicode[STIX]{x1D6FF}(r=0)=0$ . In the limit $\unicode[STIX]{x1D6FA}=0$ , the equation admits the solution $\unicode[STIX]{x1D6FF}=2\sqrt{(70/39)(r/Re)}$ , which agrees with the $\unicode[STIX]{x1D6FF}\approx \sqrt{r/Re}$ behaviour established from dimensional argument of (2.1b ).

Figure 3 illustrates the influence of rotation on the boundary-layer height. In the limit of $\unicode[STIX]{x1D6FA}=0$ , the classical boundary-layer result is recovered (Watson Reference Watson1964; Schlichtling Reference Schlichtling2000). As expected, the rotation can have a tangible effect as it tends to lower the boundary-layer height. This is clearly reflected in (2.1b ) when expression (2.8) is used. We see that the centrifugal effect, which enhances inertial effect by growing like $Re\unicode[STIX]{x1D6FA}^{2}r$ , tends to compete with the radial convective effect, which is reflected by the term $Reuu_{r}$ . An estimate of the order of magnitude of this term is reached upon taking $u$ to correspond to the free-surface value. In this case, we see that $Reuu_{r}$ decays like $Re/r$ . Thus, close to the impingement point, the convective and viscous effects must balance in (2.1b ), and the boundary-layer height grows like $\unicode[STIX]{x1D6FF}\approx \sqrt{r/Re}$ regardless of the rotation speed. This is clearly depicted in figure 2 where all curves merge for $Re^{-1/3}r<0.1$ . Far from the impingement point, the centrifugal and viscous effects must balance, and the height decays like $\unicode[STIX]{x1D6FF}\approx \sqrt{1/Re\unicode[STIX]{x1D6FA}^{2}r}$ . Therefore, rotation has a similar effect of thinning the boundary layer like slip (Prince et al. Reference Prince, Maynes and Crockett2012; Khayat Reference Khayat2016) or shear thinning (Khayat 2014). Interestingly, unlike slip flow, where the boundary-layer height invariably increases in the radial direction, a high rotation rate causes the boundary-layer height to decrease with radial distance after reaching a maximum as shown in figure 3.

A maximum in the boundary-layer height is therefore expected to emerge as a result of the growing centrifugal effect with radial distance. The maximum height $\unicode[STIX]{x1D6FF}_{max}$ coincides with the balance between the radial convective and the centrifugal effects, at a distance that is roughly dictated by $1/r\sim \unicode[STIX]{x1D6FA}^{2}r$ . Thus, the location $\unicode[STIX]{x1D70C}_{max}$ of the maximum decreases rather rapidly with rotation speed, like $1/\unicode[STIX]{x1D6FA}$ , and the maximum height behaves like $\unicode[STIX]{x1D6FA}^{-1/2}$ . Considering that the inherent reason of the boundary-layer growth is due to accumulating viscous effect, the location of the maximum boundary-layer height occurs closer to the disk centre when $\unicode[STIX]{x1D6FA}$ increases.

Figure 3.

Influence of rotation in the developing boundary-layer region (ii). The boundary-layer thickness is plotted against the radial distance for different rotation speed.

We next examine the flow downstream of the transition point.

3.2 Influence of rotation on the transition location and film thickness

As the boundary layer grows with radial distance, it eventually invades the entire film width, merging with the jet free surface at $r=r_{0}$ . For $r<r_{\text{0}}$ and above the boundary-layer outer edge, at some height $z=h(r)>\unicode[STIX]{x1D6FF}(r)$ , lies the free surface. The height of the free surface in region (ii) is then determined from mass conservation inside and outside the boundary layer. Therefore, for $r<r_{0}$ ,

(3.4) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x1D6FF}(r)}u(r,z)\,\text{d}z+h(r)-\unicode[STIX]{x1D6FF}(r)={\displaystyle \frac{1}{2r}}.\end{eqnarray}$$

Upon substituting the velocity profile (3.1), the film thickness is obtained as

(3.5) $$\begin{eqnarray}h(r<r_{0})={\displaystyle \frac{1}{48}}{\displaystyle \frac{24+18r\unicode[STIX]{x1D6FF}-Re\unicode[STIX]{x1D6FA}^{2}r^{2}\unicode[STIX]{x1D6FF}^{3}}{r}}.\end{eqnarray}$$

The term $Re\unicode[STIX]{x1D6FA}^{2}r^{2}\unicode[STIX]{x1D6FF}^{3}$ reflects the thinning effect of rotation on the film thickness. In the absence of rotation, we recall that $\unicode[STIX]{x1D6FF}=2\sqrt{(70/39)(r/Re)}$ , so that

(3.6) $$\begin{eqnarray}h(r<r_{0};\unicode[STIX]{x1D6FA}=0)={\displaystyle \frac{1}{4}}\left({\displaystyle \frac{2}{r}}+\sqrt{{\displaystyle \frac{210}{13}}{\displaystyle \frac{r}{Re}}}\right).\end{eqnarray}$$

Hence, in the absence of rotation, $h$ decreases rapidly, like $1/r$ , near the disk centre, reaching a minimum, and increases like $\sqrt{r}$ further downstream. In contrast, in the presence of strong centrifugal effect, upon recalling that $\unicode[STIX]{x1D6FF}\sim \sqrt{1/Re\unicode[STIX]{x1D6FA}^{2}r}$ , we find that the film thickness also behaves like $h(r<r_{0})\sim \sqrt{1/Re\unicode[STIX]{x1D6FA}^{2}r}$ . This will be discussed further later.

Figure 4 depicts the influence of rotation on the transition location $r_{0}$ and corresponding film thickness $h_{0}\equiv h(r_{0})$ , which are determined by setting $h(r_{0})=\unicode[STIX]{x1D6FF}(r_{0})$ . The transition location increases as the film becomes thinner with increasing rotation speed as inertia is enhanced by rotation. This is the same trend predicted for the effect of slip (see figure 8 of Khayat (Reference Khayat2016)). However, in contrast to the effect of slip, the rotation speed does not yield an asymptotic behaviour of the thickness for large $\unicode[STIX]{x1D6FA}$ . Another interesting contrasting behaviour is the slow deviation from the $\unicode[STIX]{x1D6FA}=0$ level for small $\unicode[STIX]{x1D6FA}$ in figure 4 as opposed to the strong departure for small slip.

Figure 4.

Dependence of location and film thickness at the transition point between the developing and viscous boundary-layer regions (ii) and (iii).

In the absence of rotation, upon equating  $h(r_{0};\unicode[STIX]{x1D6FA}=0)=1/4(2/r_{0}+\!\sqrt{(210/13)(r_{0}/Re)})$ and $\unicode[STIX]{x1D6FF}(r_{0};\unicode[STIX]{x1D6FA}=0)=2\sqrt{(70/39)(r_{0}/Re)}$ , we obtain

(3.7) $$\begin{eqnarray}r_{0}(\unicode[STIX]{x1D6FA}=0)=\left({\displaystyle \frac{78}{875}}Re\right)^{1/3}.\end{eqnarray}$$

In particular, $Re^{-1/3}r_{0}=0.447$ and $Re^{1/3}h(r_{0})=1.791$ are the limit values reflected in figure 4. In comparison, Watson’s transition location is given by $Re^{-1/3}r_{0}=9\sqrt{3}c(\unicode[STIX]{x03C0}-c\sqrt{3})/16\unicode[STIX]{x03C0}^{2}=0.463$ , where $c=1.402$ (Watson Reference Watson1964), which reflects a discrepancy less than 3 %.

5.1 General formulation

Using equation (2.1a ), the integral form of (2.1b ) at any location becomes

(5.1) $$\begin{eqnarray}{\displaystyle \frac{1}{r}}{\displaystyle \frac{\text{d}}{\text{d}r}}r\int _{0}^{h}(u^{2}-u)\,\text{d}z-{\displaystyle \frac{1}{r}}\int _{0}^{h}v^{2}\,\text{d}z=-{\displaystyle \frac{1}{Fr^{2}}}\int _{0}^{h}{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}r}}\,\text{d}z-{\displaystyle \frac{1}{Re}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}}\right|_{z=0}.\end{eqnarray}$$

We recall the position of the hydraulic jump as being $r=r_{j}$ , and let $h_{j}=h(r=r_{j})$ and $u_{j}(z)=u(r=r_{j},z)$ be the film thickness and radial velocity component immediately upstream of the jump (see figure 1). Downstream of the jump $(r>r_{j})$ , the film height and velocity are denoted by $H(r)$ and $U(r,z)$ respectively. Immediately downstream of the jump, we introduce $H_{j}=H(r=r_{j})$ and $U_{j}(z)=U(r=r_{j},z)$ . We observe that the azimuthal velocity component at the jump is given by $v_{j}=\unicode[STIX]{x1D6FA}r_{j}$ as per (2.8). Across the jump, equation (5.1) is applied to a control volume of width $\unicode[STIX]{x0394}r$ in the radial direction, yielding

(5.2) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{0}^{H_{j}}(U_{j}^{2}-U_{j})\,\text{d}z-\int _{0}^{h_{j}}(u_{j}^{2}-u_{j})\,\text{d}z-{\displaystyle \frac{\unicode[STIX]{x0394}r}{r_{j}}}\int _{0}^{h_{j}}v_{j}^{2}\,\text{d}z\nonumber\\ \displaystyle & & \displaystyle \qquad =-{\displaystyle \frac{1}{Fr^{2}}}\left(\int _{0}^{H_{j}}P_{j}\,\text{d}z-\int _{0}^{h_{j}}p_{j}\,\text{d}z\right)-{\displaystyle \frac{\unicode[STIX]{x0394}r}{Re}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}z}}\right|_{z=0}.\end{eqnarray}$$

Here, $p_{j}(z)$ and $P_{j}(z)$ are the pressures immediately upstream and downstream of the jump. Since the width of the jump $\unicode[STIX]{x0394}r$ is assumed to be small, the terms in (5.2) containing $\unicode[STIX]{x0394}r$ become negligible. In this case, we recall the pressure terms from (2.6), and use (2.9) for mass conservation. In addition, following Bush & Aristoff (Reference Bush and Aristoff2003), we include the effect of surface tension. The resulting equation takes the same form as in the case of a stationary disk, namely

(5.3) $$\begin{eqnarray}{\displaystyle \frac{1}{2}}(H_{j}^{2}-h_{j}^{2})+{\displaystyle \frac{1}{Bo}}{\displaystyle \frac{(H_{j}-h_{j})}{r_{j}}}=Fr^{2}\int _{0}^{h_{j}}u_{j}^{2}\,\text{d}z-Fr^{2}\int _{0}^{H_{j}}U_{j}^{2}\,\text{d}z.\end{eqnarray}$$

We observe that the effect of rotation is reflected implicitly in (5.3) in the height and velocity upstream and downstream of the jump. As in the case of a stationary disk, some assumptions are made regarding the nature of the flow downstream of the jump.

In this work, we follow Duchesne et al. (Reference Duchesne, Lebon and Limat2014), and adopt a lubrication flow assumption in the presence of rotation. In this case, a differential equation for $H$ can be obtained by neglecting the first two inertia terms in (2.1b ). In other words, we assume $Re$ to be small but $Re\unicode[STIX]{x1D6FA}^{2}$ non-negligible. We note that the hydrostatic pressure term is not negligible due to the large film thickness downstream of the jump. Consequently, the remaining terms in (2.1b ), the hydrostatic pressure, the viscous and the centrifugal effects, must balance:

(5.4) $$\begin{eqnarray}Re\unicode[STIX]{x1D6FA}^{2}r=G{\displaystyle \frac{\text{d}H}{\text{d}r}}-U_{zz}.\end{eqnarray}$$

By applying the no-slip condition at the fluid–disk interface and the no-shear condition at the free surface, (5.4) can be integrated to yield the velocity downstream of the jump:

(5.5) $$\begin{eqnarray}U(r,z)=Re\left({\displaystyle \frac{1}{Fr^{2}}}{\displaystyle \frac{\text{d}H}{\text{d}r}}-\unicode[STIX]{x1D6FA}^{2}r\right)\left({\displaystyle \frac{z^{2}}{2}}-Hz\right).\end{eqnarray}$$

Upon inserting $U$ into the mass conservation equation (2.9) and integrating, the governing equation for the film thickness downstream of the jump becomes

(5.6) $$\begin{eqnarray}{\displaystyle \frac{\text{d}H}{\text{d}r}}=-{\displaystyle \frac{3}{2}}{\displaystyle \frac{Fr^{2}}{Re}}{\displaystyle \frac{H^{-3}}{r}}+Fr^{2}\unicode[STIX]{x1D6FA}^{2}r.\end{eqnarray}$$

The corresponding velocity reduces to

(5.7) $$\begin{eqnarray}U(r,z)=-{\displaystyle \frac{3}{2r}}H^{-3}\left({\displaystyle \frac{z^{2}}{2}}-Hz\right).\end{eqnarray}$$

The problem remains open as to the existence of an additional relation, which would allow us to determine uniquely the height and the location of the jump. This additional relation can be based on a boundary condition for the film height specified at the edge of the disk, that is $H(r=r_{\infty })=H_{\infty }$ . However, the specification of the height at the edge is theoretically not a simple matter. We discuss this issue by revisiting first the flow over a stationary disk.

5.2 The hydraulic jump over a stationary disk

In the absence of disk rotation, and subject to $H(r=r_{\infty })=H_{\infty }$ , an analytical solution exists for (5.6):

(5.8) $$\begin{eqnarray}H(r)=\left(H_{\infty }^{4}+6{\displaystyle \frac{Fr^{2}}{Re}}\ln \left({\displaystyle \frac{r_{\infty }}{r}}\right)\right)^{1/4}.\end{eqnarray}$$

Obviously, in the presence of radial flow, the film thickness at the edge of the disk cannot be zero. Direct measurements by Duchesne et al. (Reference Duchesne, Lebon and Limat2014) of this edge thickness, performed at nearly 5 mm of the disk perimeter in their experiment, give a nearly constant value of 3 mm with a weak power-law variation not exceeding a few per cent. This constant thickness value is very close to the capillary length $\sqrt{\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}g}$ of the fluid, which results from the balance of forces between the hydrostatic pressure and the surface tension at the disk perimeter. This value is also consistent with the measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010). Similarly, we assume here that the film thickness at the edge of the disk is essentially equal to the film thickness the liquid exhibits under static conditions. This is not an unreasonable assumption since the flow downstream of the jump has predominantly the character of gradually varied flow. This static thickness is governed by the minimum free energy, and was given by Lubarda & Talke (Reference Lubarda and Talke2011) as $h_{s}=2\sqrt{\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}g}\sin (\unicode[STIX]{x1D703}_{Y}/2)$ , where $\unicode[STIX]{x1D703}_{Y}$ is the contact angle.

In addition, in order to explore the small variation of edge thickness with flow rate as observed by Duchesne et al. (Reference Duchesne, Lebon and Limat2014), we resort to a minimum mechanical energy principle (Yang & Chen Reference Yang and Chen1992; Yang, Chen & Hsu Reference Yang, Chen and Hsu1997). This principle states that a fluid flowing over the edge of a disk under the influence of a hydrostatic pressure gradient will adjust itself so that the mechanical energy within the fluid will be minimum with respect to the film thickness at the disk edge. We note that this approach originates from open-channel hydraulics (Bakhmeteff Reference Bakhmeteff1966). Consequently, the contribution on the thickness at the edge of the disk can be calculated by setting the derivative of the mechanical energy with respect to the film thickness equal to zero, that is

(5.9) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}H}}\int _{0}^{H}\left(Fr^{2}{\displaystyle \frac{U^{2}}{2}}+H\right)\,\text{d}z=0.\end{eqnarray}$$

After substituting for $U$ from (5.7) into (5.9), we obtain the dependence of the film thickness on the Froude number (flow rate) at the edge of the disk: $(3/40)^{1/3}(Fr/r_{\infty })^{2/3}$ .

Taking into consideration this small dependence, we obtain the dimensionless film thickness at the edge as

(5.10) $$\begin{eqnarray}H_{\infty }=2\sqrt{{\displaystyle \frac{1}{Bo}}}\sin \left({\displaystyle \frac{\unicode[STIX]{x1D703}_{Y}}{2}}\right)+\left({\displaystyle \frac{3}{40}}\right)^{1/3}\left({\displaystyle \frac{Fr}{r_{\infty }}}\right)^{2/3}.\end{eqnarray}$$

Upon specifying the thickness $H(r=r_{\infty })=H_{\infty }$ at the edge of the disk as per (5.10), we obtain the film thickness distribution downstream of the jump from (5.8), which is then substituted into the momentum balance equation (5.3) to determine the location of the jump. The value of $\unicode[STIX]{x1D703}_{Y}$ may then be adjusted from experiment. For the current problem, we achieve a good agreement with the measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) by taking $\unicode[STIX]{x1D703}_{Y}=90^{\circ }$ . This value is within the range of values measured for water on polydimethylsiloxane (Diversified Enterprises 2009), which is the material used by Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) for their disk.

The dependence of the jump location on the Froude number is reported in figure 13, where a comparison with experiment is presented based on the data of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) for water, leading to good agreement. The original experimental data were reported in terms of the flow rate for a jet radius of 1 mm and a speed ranging approximately from 3 to $16~\text{m}~\text{s}^{-1}$ . In this case, taking $g\simeq 10~\text{m}~\text{s}^{-2}$ , $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ and $\unicode[STIX]{x1D6FE}=70~\text{dyn}~\text{cm}^{-1}$ , we have $Bo=0.13$ and $G=10^{-2}$ .

Figure 13.

Dependence of hydraulic jump location on flow rate (Froude number) for a stationary disk. The figure shows the comparison between the current theoretical predictions and the measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) for a water jet.

Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) and Rojas, Argentina & Tirapegui (Reference Rojas, Argentina and Tirapegui2013) found that the jump radius scales approximately as $Q^{5/8}\unicode[STIX]{x1D708}^{-3/8}g^{-1/8}$ . Avedisian & Zhao (Reference Avedisian and Zhao2000) investigated the circular hydraulic jump experimentally for normal and reduced gravity conditions. They measured the jump diameter and shape at the free liquid surface for an impinging jet on a stationary disk. Based on the reported two values of the flow rate and two gravitational acceleration data provided, we find that the location of the jump behaves close to $g^{-1/9}$ , roughly confirming the scaling of Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) for low gravity. Using the current scaling, the dimensionless form of Bohr et al. estimate can be written in terms of the transition location and the Froude number as $r_{j}\sim r_{0}^{9/8}Fr^{1/4}$ , where we recall $r_{0}(\unicode[STIX]{x1D6FA}=0)=(78/875Re)^{1/3}$ from (3.7). More recently, Duchesne et al. (Reference Duchesne, Lebon and Limat2014) derived a more accurate estimate involving a logarithmic correction based on the thin-film approach. When cast in dimensionless form, we find that the jump location behaves like $r_{j}\sim Fr_{j}^{-1}Fr^{5/8}G^{-3/8}$ , where $Fr_{j}$ is the Froude number defined in terms of the jump height and the average velocity immediately after the jump. In our notation, $Fr_{j}=Fr/2r_{j}H_{j}^{3/2}$ . Duchesne et al. (Reference Duchesne, Lebon and Limat2014) found that $Fr_{j}$ remains sensibly independent of the flow rate (constant with respect to $Fr$ ), especially for a jet of fluid of low viscosity or at relatively large flow rate. Figure 14 shows the behaviour of $Fr_{j}$ against $Fr$ , corresponding to the prediction in figure 13, based on the current formulation. It is found that $Fr_{j}$ remains sensibly equal to 0.15. As a way to validate the current approach based on the cubic velocity profile, we also included in figure 14 the result based on Watson’s similarity solution, which indicates that both approaches yield essentially the same value for $Fr_{j}$ . As a reference, we also reproduced the measurements of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) for silicon oil. In this case, it turns out that we essentially have $Fr_{j}\simeq 0.33$ . However, the data also suggest a deviation from the constant level at very flow rate. This issue is further explored next by comparing the location of the jump in the low flow rate range.

It appears that in order to determine both the location and the height of the jump, we seem to have two alternatives: fixing the value of $Fr_{j}$ or imposing the film thickness at the edge. Based on the results reported in figures 13 and 14, the two alternatives are essentially equivalent. It is therefore important to further examine conditions where $Fr_{j}$ may not remain independent of the flow rate. This is likely the case for impinging flow at low flow rate or a fluid of small viscosity. For this purpose, we conduct a comparison against the data of Hansen et al. (Reference Hansen, Horluck, Zauner, Dimon, Ellegaard and Creagh1997), who measured the position of the hydraulic jump against the flow rate, ranging from 0 to $60~\text{cm}^{3}~\text{s}^{-1}$ . In this case, the contact angle in (5.10) was taken for water and glass as $35^{\circ }$ (Vicente, Andre & Ferreira Reference Vicente, Andre and Ferreira2012). Figure 15 depicts the comparison between our formulation and experiment. The experimental data are reproduced in dimensionless form. The distribution of $Fr_{j}$ versus $Fr$ is also included for reference. The figure clearly confirms that $Fr_{j}$ remains constant over the large Froude number (flow rate) range. At small flow rate, $Fr_{j}$ increases rather rapidly as $Fr$ tends to zero.

Figure 14.

Dependence of $Fr_{j}$ on the flow rate (Froude number) for a stationary disk. The solid and dashed lines show the results corresponding to the configuration of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) for water, based on the current cubic and similarity profiles, respectively. The data corresponding to silicon oil from Duchesne et al. (Reference Duchesne, Lebon and Limat2014) are also included.

It is clear from figures 13 and 15 that the range of Froude numbers is not the same. The data of Hansen et al. (Reference Hansen, Horluck, Zauner, Dimon, Ellegaard and Creagh1997) are restricted to the very low range with no overlap with the data of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010). Therefore, we confirm from the observations of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) as well as from their data in figure 14, that $Fr_{j}$ is not expected to remain constant in the low Froude number range.

Figure 15.

Dependence of hydraulic jump location on flow rate (Froude number) for a stationary disk. The figure shows the comparison between the current theoretical predictions and the measurements of Hansen et al. (Reference Hansen, Horluck, Zauner, Dimon, Ellegaard and Creagh1997) for a water jet on a glass disk. The distribution of $Fr_{j}$ is also included.

Finally, we examine the validity of our approach regarding the actual size and shape of the jump. In their experiment of the flow of a thin film, Ozar et al. (Reference Ozar, Cetegen and Faghri2003) measured the jump profile against the radial distance on a stationary disk. The comparison between theory and experiment is reported in figure 16 for two different flow rates for a disk of radius 203 mm or $r_{\infty }=4$ . In this case, we take the contact angle for water and aluminium to be $70^{\circ }$ (Majeed Reference Majeed2014). For $Q=7~\text{l}~\text{mn}^{-1}$ , figure 16(a) shows that the jump is located far upstream of the disk edge $(r\approx 2)$ , whereas for $Q=15~\text{l}~\text{mn}^{-1}$ , figure 16(b) shows that the jump is located close to the edge. In both cases, the jump height exhibits a maximum and tapers rather rapidly near the edge. The comparison shows a close agreement for the thickness distribution upstream of the jump as well as for the (average) location and height of the jump for the two flow rates.

Figure 16.

Comparison between theory and the measurements of Ozar et al. (Reference Ozar, Cetegen and Faghri2003) for the hydraulic jump of a thin film flowing on a stationary disk for $Q=7~\text{l}~\text{mn}^{-1}$ (a) and $Q=15~\text{l}~\text{mn}^{-1}$ (b).

5.3 The hydraulic jump over a rotating disk

In the case of a rotating disk, the thickness at the edge is expected to vary with the speed of rotation as well as the jet flow rate and surface tension. In this case, while the minimum energy principle is expected to hold, the static film thickness must be adjusted to account for rotation. We expect the thickness at the edge to decrease with rotation speed and increase with the flow rate as it should somehow reflect a balance between the centrifugal acceleration, the hydrostatic pressure and the surface tension. However, the measurements of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) as well as those of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) seem to suggest little influence of the flow rate, which is also supported by our predictions in § 5.2 above. Although these observations are conclusive for a stationary disk, we expect them to hold for a rotating disk.

However, the influence of rotation speed on the thickness at the edge of a rotating disk remains unaddressed. Here, we examine the effect of rotation by taking guidance from the extensive existing insight in thin-film spin coating. In this process, a drop of liquid solution is deposited onto a rotating substrate (wafer). After an initial acceleration, the liquid reaches a uniform thickness that is sufficiently small for the viscous shear drag to balance the centrifugal acceleration. At this stage, the film begins to dry up as the solvent evaporates, ultimately exhibiting a constant and steady thickness. Theory as well as experiment suggest that the final thickness varies with rotation speed like $h_{r}\sim \unicode[STIX]{x1D708}^{1/3}\unicode[STIX]{x1D714}^{-2/3}$ (Hall, Underhill & Torkelson Reference Hall, Underhill and Torkelson1998), where the kinematic viscosity $\unicode[STIX]{x1D708}$ appears to be the main influencing liquid property. This behaviour is observed at relatively large spin speed, and does not take into account the static limit that corresponds to $\unicode[STIX]{x1D714}=0$ . We find the overall behaviour that seems to fit well experiment to be of the form:

(5.11) $$\begin{eqnarray}h_{r}={\displaystyle \frac{h_{s}}{1+\unicode[STIX]{x1D6FC}h_{s}\left({\displaystyle \frac{\unicode[STIX]{x1D714}^{2}}{\unicode[STIX]{x1D708}}}\right)^{1/3}}}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D6FC}$ is an empirical coefficient in units of $(\text{s}~\text{m})^{-1/3}$ . To illustrate the applicability of (5.11), we examine the flow of photoresist AZ6600 liquid series, which are liquid films used as resistant substance applied as a coating to protect a surface during some process, for example to prevent dye or glaze adhering (MicroChemicals 2013). Figure 17 shows the data and curves based on (5.11) for five AZ6600 liquid films based on measurements reported by Clariant GmbH (see reference source below). In this case, all five liquid films have sensibly the same surface tension for a contact angle of $90^{\circ }$ , namely $10~\text{dyn}~\text{cm}^{-2}$ (Bauer et al. Reference Bauer, Drescher, Silz, Frankenfeld and Illig1997), yielding $h_{s}=1.7~\text{mm}$ . The kinematic viscosities are as follows: 19 cSt (AZ6612), 27.7 cSt (AZ6615), 34.3 cSt (AZ6618), 58.5 cSt (AZ6624) and 82 cSt (AZ6632). Figure 17 shows a significant drop in the film thickness from the static level at low spin speed, reflecting the difficulty of the liquid to spread initially. Figure 17 shows the final film thickness as function of rotation speed in the spin coating of AZ6600 photoresist liquid films. The values of $\unicode[STIX]{x1D6FC}$ are given for a best fit of expression (5.11). The data are reproduced from MicroChemicals (2013). The inset shows data amplified in the small thickness range.

Figure 17.

Final film thickness as function of rotation speed in the spin coating of AZ6600 photoresist liquid films. The values of $\unicode[STIX]{x1D6FC}$ are given for a best fit of expression (5.11). Data are reproduced from MicroChemicals (2013). Inset shows data amplified in the small thickness range.

Figure 18.

Influence of rotation on the hydraulic jump location, plotted against the Froude number (flow rate) for the AZ6612 photoresist fluid.

We now illustrate the influence of rotation on the jet flow and hydraulic jump by examining the behaviour of the AZ6612 liquid. The flow configuration is the same as that of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) discussed in § 5.2 for the stationary disk; only the fluid is now AZ6612 instead of water and the jet radius is 4 mm instead of 1 mm. The location of the jump and corresponding height are shown respectively in figures 18 and 19 against the Froude number (flow rate). We can see that the location of the jump is pushed outwards with higher rotating speed as well as higher flow rate, which is consistent with existing experimental findings (Ozar et al. Reference Ozar, Cetegen and Faghri2003; Deng & Ouyang Reference Deng and Ouyang2011). Here, the Coriolis acceleration associated with rotation can be seen as weakening the effect of gravity. Indeed, an interesting parallel can be established between rotation and body forces. The measurements of Avedisian & Zhao (Reference Avedisian and Zhao2000) show that the steady state diameter of the jump under reduced gravity is larger than the diameter of the same jump under normal gravity conditions. Similar to the current case of a rotating disk, they predicted that the hydraulic jump would also disappear in the absence of gravity.

For a stationary disk $(\unicode[STIX]{x1D6FA}=0)$ , figure 18 suggests that the location of the jump grows essentially linearly with flow rate. On a rotating disk, the growth rate increases for large rotating speeds.

The jump height in figure 19 is shown to grow monotonically with flow rate for a stationary disk, but tends to exhibit a maximum at low rotation speed. At higher rotation speed, the jump actually decreases with flow rate, suggesting that the jump weakens as it is being pushed towards the disk edge. Although there is no indication from the figure that the jump has reached the edge of the disk ( $r_{\infty }=90$ ), this may not be the case in reality. A more accurate prediction would have to be based on a formulation that includes inertia since convective effects are bound to become more significant at higher flow rate.

Figure 19.

Influence of rotation on the hydraulic jump height, plotted against the Froude number (flow rate) for the AZ6612 photoresist fluid.

The overall effect of rotation is illustrated in figures 20 and 21, which depict the film thickness (figure 20) and the corresponding surface velocity (figure 21) over the entire disk for $Fr=30$ . The jump height decreases with increasing rotation rate, simultaneously as the jump location is pushed outwards toward the edge of the disk. However, the film thickness and surface velocity upstream of the jump do not seem to vary significantly with $\unicode[STIX]{x1D6FA}$ over range reported here.

Figure 20.

Influence of rotation on the film thickness plotted against radial distance in regions (ii), (iii) and (iv) for the AZ6612 fluid.

Overall, the velocity in figure 21 is much less influenced by the rotation speed than the film thickness. For a stationary and low rotation speed $(\unicode[STIX]{x1D6FA}=0.001)$ , the existence of the jump is unquestionable. However, with higher rotation speed, the jump becomes weaker, which may signal the beginning of the vanishing of hydraulic jump. Both the thickness and velocity decrease monotonically with radial distance, followed by a drastic thinning near the edge of the disk that is enhanced by centrifugal effect.

Figure 21.

Influence of rotation on the film free-surface velocity plotted against radial distance in regions (ii), (iii) and (iv) for the AZ6612 fluid. Inset shows the amplification around the jump.

Finally, we observe from figures 20 and 21 that the flow appears insensitive to the variation of the rotation speed. This insensitivity corroborates well with experiment at least for the range of speeds reported here.